On Hereditary, Semihereditary and Quasi-Hereditary Ternary Rings

Abstract

In this paper, we define right hereditary, semihereditary and quasi-hereditary ternary rings, as previously introduced in binary rings. We show that, if a ternary ring T is completely reducible, then every right ideal I of T is of the form I = e.1.T, where e is an idempotent. Consequently, T is a right hereditary ternary ring. We also prove that, if a reduced ternary ring T satisfies the minimal condition on right annihilators of idempotents, and if I= 0 is a right ideal satisfying the condition: (for every right ideal K ⊂ I, there exists a minimal right ideal H of T such that H ⊂ K), then I is projective as a right T–module. Finally, we show that if T is a semiprimary ternary ring in which the Jacobson radical = {0}, then T is a right hereditary and quasi-hereditary ternary ring.

Key words and phrases. Hereditary ternary ring; Semihereditary ternary ring; Quasi-hereditary ternary ring.

2020 Mathematics Subject Classification. 20N10